Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51 403 -0.0012 -0.0009 -0.0006 -0.0003 0.0000 1 2 3 4 5 Shear strain γ rθ δ = E / E' ϕ o =0 o , 90 o 15 o 30 o 45 o 60 o 75 o γ rθ Figure 3 : The dependence of the shear strain γ rθ on the anisotropy ratio δ for a series of ϕ o -values. in the case of “transtropic” discs γ rθ is non-zero (excluding the limiting values ϕ o =0 o , ϕ o =90 o , for which symmetry imposes zeroing of γ rθ ) and the deformation in the immediate vicinity of the disc center becomes both distortive and dilatational rather than purely dilatational. The magnitude of γ rθ increases constantly with increasing δ for ϕ o -values in the 0 o < ϕ o <45 o range. On the contrary, for ϕ o -values in the 45 o < ϕ o <90 o range γ rθ increases for low δ -values and then it reaches a plateau. The dependence of the magnitude of γ rθ on the angle ϕ o is non-monotonous exhibiting its maximum value around ϕ o ≈30 o . For the differences of the deformation between an isotropic and a “transtropic” disc to become more evident, and in order to highlight the loss of some symmetries which are “familiar” in case of isotropic materials, the deformed shape of an element (Δ r , Δ θ )=(15π R /180, 15π R /180) of the disc cross-section, oriented along the loading line, is drawn in Figs.4(a, b). This element is located in the immediate vicinity of the disc-jaw contact arc, i.e., in the area where the deformation field is extremely amplified due to the “direct” action of the external loading scheme (see the sketches embedded in Figs.4). Both the undeformed element (black dashed line) and the respective deformed ones are drawn: The deformed element for the isotropic configuration is drawn with green line while that for the “transtropic” one with red line. Two geometries were con- sidered, one for ϕ o =20 o (Fig.4a) and a second one for ϕ o =60 o (Fig.4b). The materials considered correspond to rocks with relatively low stiffness in order for the deformation to be clearly visible. For the “fictitious” isotropic material it is assumed that E = E ΄=4 GPa, ν = ν ΄=0.34 while for the “fictitious” “trans- tropic” one it is assumed that E =4 GPa, ν =0.34, E ΄=2 GPa, ν ΄= ν yx =0.2 (and it follows that ν xy =0.40<0.50). Again for clarity reasons the value of the load imposed is rather “unrealistic” ( P frame =100 kN). Given that the analysis introduced is linearly elastic, it is obvious that the magnitude of the load imposed will only quantitative influence the results; from a qualitative point of view the conclusions drawn are not influenced by the specific parameter. Finally, it is mentioned that the length of the loading arc (2 ω o ) is not constant but rather it is obtained with the aid of the solution of the respective isotropic disc-jaw contact problem. As a result it depends both on the inclination of the loading axis with respect to the anisotropy planes, ϕ o , and also on the properties of the disc materials. In this context, for the con- figuration with ϕ o =20 o (Fig.4a) it is calculated that ω ο ( ϕ o =20 o )=23.05 o for the “fictitious” “transtropic” disc while for the same configuration for the “fictitious” isotropic material it is calculated that ω ο ( ϕ o =20 o )=24.52 o . Similarly for the configu- ration with ϕ o =60 o (Fig.4b) it is determined that ω ο ( ϕ o =60 o )=22.29 o for the “fictitious” “transtropic” disc and ω ο ( ϕ o =60 o )= 24.52 o for the “fictitious” isotropic disc (obviously for the isotropic disc the length of the contact arc does not depend on the inclination angle ϕ o of the loading axis with respect to x -axis of the reference system (see Fig.1)). The most important conclusion drawn from Figs.4(a, b) (besides the expected differences between the “transtropic” and the isotropic disc in the “rigid body” translation and rotation of the element in question, as intermediate steps from the un- strained to its final strained state), is the evident asymmetry detected in the variation of shear deformation in case of “trans- tropic” materials, at symmetric points, i.e., of the corner points of the element in question, even when located along the loaded radius. Indeed, while for the isotropic material γ rθ ( r , θ )= γ rθ ( r , θ +Δ θ ) all along the loaded radius, this is by no means